Abstract
In the magnetic diffusion problem, a magnetic diffusion equation is coupled by an Ohmic heating energy equation. The Ohmic heating can make the magnetic diffusion coefficient (i.e. the resistivity) vary violently, and make the diffusion a highly nonlinear process. For this reason, the problem is normally very hard to be solved analytically. In this article, under the condition of a step-function resistivity and a constant boundary magnetic field, we successfully derived an exact solution for this nonlinear problem. The solution takes four parameters as input: the fixed magnetic boundary \(B_0\), \(\eta _{\text {S}}\) and \(\eta _{\text {L}}\) that are resistivities below and above the critical energy density of a material, and the critical energy density \(e_{\text {c}}\) of the material. The solution curve B(x, t) possesses the characteristic of a sharp front, and its evolution obeys the usual self-similar rule with the similarity variable \(x/\sqrt{t}\).
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Data Availability Statement
The manuscript has associated data in a data repository. [Authors’ comment: The data that support the findings of this study are available from the corresponding author upon reasonable request].
Notes
This conclusion is in fact just a different way of expression to the well-known property for the self-similar solution of a usual diffusion equation, where it is usually stated like “the self-similarity variable is \(x/\sqrt{t}\)”.
It can be seen from Eq. (25) that the integration in Eq. (16) is finally independent of x. This fact is a key for the existence of a self-similarity solution for the magnetic diffusion coupled by Ohmic heating. (This footnote can be taken as a response to the discussions about self-similarity on the head of Sec. 3).
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Acknowledgements
Shu was partially supported by the National Natural Science Foundation of China (No. 12371373). Feng and Zhao were partially supported by the National Key Research and Development Program of China (Grant No. 2023YFB3001604), the National Natural Science Foundation of China (No. 12371373) and the GHfund A (202302011094).
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Xiao, B., Wang, G., Zhao, L. et al. An exact solution for the magnetic diffusion problem with a step-function resistivity model. Eur. Phys. J. Plus 139, 305 (2024). https://doi.org/10.1140/epjp/s13360-024-05086-2
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DOI: https://doi.org/10.1140/epjp/s13360-024-05086-2